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arXiv · 2110.05277

On the energy-critical quadratic nonlinear Schr\"odinger system with three waves

Fanfei Meng·Sheng Wang·Chengbin Xu
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Abstract

In this article, we consider the dynamics of the energy-critical quadratic nonlinear Schr\"odinger system $\[ \left\{ \begin{aligned} & i u^1_t + \kappa_1 \Delta u^1 = -\overline{u^2}u^3, \\ & i u^2_t + \kappa_2 \Delta u^2 = -\overline{u^1}u^3, \\ & i u^3_t + \kappa_3 \Delta u^3 = -u^1u^2, \\ \end{aligned} \right. \qquad (t, x) \in \R \times \R^6 \] in energy-space $ {\dot H}^1 \times {\dot H}^1\times{\dot H}^1 $, where the sign of potential energy can not be determined. We prove the scattering theory with mass-resonance (or with radial initial data) below ground state via concentration compactness method. We discover a family of new physically conserved quantities with mass-resonance which play an important role in the proof of scattering.

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Fanfei Meng, Sheng Wang, Chengbin Xu. 2021-10-11. On the energy-critical quadratic nonlinear Schr\"odinger system with three waves. https://arxiv.org/abs/2110.05277

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